Nonlinear space-time dynamics, defined in terms of celebrated `solitonic` equations, brings indispensable tools for understanding, prediction and control of complex behaviors in both physical and life sciences. In this paper, we review sine--Gordon solitons, kinks and breathers as models of nonlinear excitations in complex systems in physics and in living cellular structures, both intra--cellular (DNA, protein folding and microtubules) and inter--cellular (neural impulses and muscular contractions).

The Lax representations for the soliton equations with $\mathbb{Z}_h$ and $\mathbb{D}_h$ reductions are analyzed. Their recursion operators are shown to possess factorization properties due to the grading in the relevant Lie algebra. We show that with each simple Lie algebra one can relate $r$ fundamental recursion operators ${\boldsymbol \Lambda}_{m_k}$ and a master recursion operator ${\boldsymbol \Lambda}$ generating NLEEs of MKdV type and their Hamiltonian hierarchies. The Wronskian relations are formulated and shown to provide the tools to understand the inverse scattering method as a generalized Fourier transform. They are also used to analyze the conservation laws of the above mentioned soliton equations.

We provide conditions for a lattice scheme defined on a four points lattice to be linearizable by a point transformation. We apply the obtained conditions to a symmetry preserving difference scheme for the Burgers potential introduced by Dorodnitsyn and show that it is not linearizable.

Completely integrable systems always admit more alternative Hamiltonian descriptions. The geometrical formulation of quantum systems shows that similar conclusions hold true also for quantum systems. In addition, the description of quantum systems on Hilbert manifolds, e.g., the complex projective space, shows that not only quantum systems admit alternative Hamiltonian descriptions, they also admit alternative linear descriptions.

This paper deals with classical solutions of the Dirichlet-Ventcel boundary value problem (BVP) for the Laplace operator in bounded sector in the plane having opening of the corresponding angle $ \varphi_{0} > 0 $. Ventcel BVP is given by second order differential operator on the boundary satisfying Lopatinksii condition there. As the boundary is non smooth, two different cases appear: $ \frac{\pi}{\varphi_{0}} $ is irrational and $ \frac{\pi}{\varphi_{0}} $ is an integer. At first we prove uniqueness result via the maximum principle and then existence of the classical solution. To do this we apply two different approaches: the machinery of the small denominators and the concept of Green function.